Use of an efficient unbiased estimator for finite population mean

In this study, we propose an improved unbiased estimator in estimating the finite population mean using a single auxiliary variable and rank of the auxiliary variable by adopting the Hartley-Ross procedure when some parameters of the auxiliary variable are known. Expressions for the bias and mean square error or variance of the estimators are obtained up to the first order of approximation. Four real data sets are used to observe the performances of the estimators and to support the theoretical findings. It turns out that the proposed unbiased estimator outperforms as compared to all other considered estimators. It is also observed that using conventional measures have significant contributions in achieving the efficiency of the estimators.


Introduction
In literature, many researchers have constructed or modified several forms of ratio, product, and regression type estimators by using the auxiliary information in estimating the finite population mean. The auxiliary information can be used either at survey stage or designing stage or estimation stage or at all stages to enhance the precision of the estimators by taking the advantage of correlation between the study variable and the auxiliary variable. In this study, we use the auxiliary variable as well as rank of the auxiliary variable at estimation stage to estimate the finite population mean. [1] were the pioneer whom used the idea of ratio of the study variable and the auxiliary variable in estimating the population mean. Singh and Singh [2] suggested the [1] type estimator when some parameters of the auxiliary variable are known in advance. [3] slightly modified the idea of [1] and suggested a new estimator for estimating the population mean. [4] used the known population parameters of the auxiliary variable in their suggested estimator for mean estimation. [5] extended the [1] estimator by using two auxiliary variables to estimate the population mean. [6,7] modified the [1] type estimator for mean estimation in simple and stratified sampling. [8] have given justification in their proposed estimator by using dual use of the auxiliary variable in their study. [9] used the dual auxiliary variable in estimating the mean of the sensitive variable under randomized response technique (RRT). [10] modified the existing ratio estimator by using the dual auxiliary information for mean estimation. [11] suggested a difference type exponential estimator based on dual auxiliary a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 variable for mean estimation. Recently [12] suggested a difference type estimator using the dual auxiliary variable under non-response in simple random sampling.
There are several estimators exist in literature which give the biased results and consequently variance or MSE tend to be inflated. This serious drawback encouraged us to construct the unbiased estimator which should be better than other considered estimators in literature. So combining the ideas of [1] and [2], we suggest an improved unbiased estimator for estimating the finite population mean.
In Section 2, we introduce some useful notations and symbols. Section 3 gives the existing estimators in literature. The proposed estimator is discussed in Section 4. The numerical results based on four real data sets are mentioned in Section 5. The conclusion is given in Section 6.

Symbols and notations
Consider a finite population Λ = Λ{Λ 1 , Λ 2 , . . ., Λ N } of N units. A simple random sample without replacement (SRSWOR) is used to draw a sample of size n units from a population. Let y i , x i , and r i be the observed values of the study variable (Y), the auxiliary variable (X) and rank of the auxiliary variable (R) respectively.
x i =n and � r ¼ P n i¼1 r i =n respectively be the sample means corresponding to the population means ðx i À � xÞ 2 =ðn À 1Þ, and ðr i À � rÞ 2 =ðn À 1Þ respectively be the sample variances corresponding to population R be the coefficients of variation of Y, X, and R respectively. Let ρ yx = S yx / (S y S x ), ρ yr = S yr / (S y S r ), and ρ xr = S xr / (S x S r ), be the correlation coefficients between their respective subscripts, where Þ=ðN À 1Þ, Þ=ðN À 1Þ, and S xr ¼ Þ=ðN À 1Þ be the population covariances between their respective subscripts. Corresponding sample covariances are Þ=ðn À 1Þ, and We define the following relative error terms to derive bias and MSE or variance expressions. Let r yr . where C yx = ρ yx C y C x , C yr = ρ yr C y C r , C xr = ρ xr C x C r , D abc ¼ m abc

Existing estimators
Now we discuss some well-known estimators in estimating the finite population mean.
1. The usual sample mean estimator is � y ð0Þ ¼ � y, and its variance, is given by 2. A general class of Hartley-Ross unbiased type estimators, is given by where � k ðjÞ i ; j = 0, 1, 2; c and d are the known population parameters of the auxiliary variable which may be coefficient of variation (C x ), coefficient of skewness (β 1x ), coefficient of kurtosis (β 2x ) and correlation coefficient (ρ yx ).
Using the assumption nðNÀ 1Þ NðnÀ 1Þ � 1 and � Y ðjÞ À � K ðjÞ � X ðjÞ ð Þ � 0, an unbiased general estimator is given by The variance of � y ðUÞ ðGÞ , is given by Note: we get the usual Hartley-Ross estimator and its variance as: and Varð� y ðUÞ , we get the [4] estimator with its variance, are given by: and Varð� y ðUÞ we get another [4] estimator and its variance, is given by: and Varð� y ðUÞ 4. A difference type estimator using a single auxiliary variable with its ranks, is given by: where d i (i = 1, 2) are constants. The variance of � y ðUÞ ðDÞ , is given by The minimum variance of � y ðUÞ ðDÞ , at optimum values of where 5.
[6] suggested the following unbiased estimator using the single auxiliary variable and is given by: where c and d are defined earlier i.e. t = −1, 0, 1; α = 0, 1 and Q is a constant whose value is to be estimated. For α = t = 1, the above estimator becomes: The minimum variance of � y ðUÞ ðCKÞ at optimum values of Q i.e. Q opt ¼ À r 2 r 1 , is given by where r 1 = r 1a + r 1b , 6. [8] suggested an idea of using rank of the auxiliary variable in the following estimator, is given by where H i (i = 1,2,3) are constants; c and d are defined earlier.
The bias of the estimator � y ðHÞ , is given by Since � X and � R are known, so replacing � Y and C yx by their consistent estimates � y and (16), the estimated bias of � y ðHÞ becomeŝ Bð� y ðHÞ Þ � ðH 1 À 1Þ� y þ H 1 � yU SubtractingBð� y ðHÞ Þ from � y ðHÞ , we get an unbiased estimator by replacing H i (i = 1,2,3) by L i (i = 1,2,3) which is considered by [10] as: � y ðUÞ ðIÞ ¼ � y ðHÞ ÀBð� y ðHÞ Þ ð18Þ or � y ðUÞ Rewriting in terms of errors, we have Solving (20), the bias of � y ðUÞ ðIÞ becomes zero. Now squaring and taking expectation of (20), the variance of � y ðUÞ ðIÞ becomes: where The minimum variance of � y ðUÞ ðIÞ is given by where

Proposed almost unbiased estimator
On the lines of [1,8,10], we propose the following alternative new unbiased estimator. This estimator is based on usual ratio, difference, and exponential ratio type estimators. The purpose is to construct an unbiased estimator that should be better than all considered estimators in estimating the finite population mean.

Numerical example
We use the following 4 real data sets for a numerical study.   The results based on Populations 1-4 are given in Tables 1-4. Tables 1-4 give the results when no conventional measures and conventional measures are used. We use the following

Conclusion
In this study, we have proposed an unbiased class of estimators in estimating the finite population mean in simple random sampling using the single auxiliary variable and rank of the auxiliary variable. Expressions for biases and MSEs or variances are obtained up to first order of approximation. Four data sets are used for numerical study. The proposed estimator outperforms in all four populations as compared to all considered estimators. It is observed that use of conventional measures i.e. C x , β 1x , β 2x , and ρ yx have significant role in increasing the efficiency of the estimators in Tables 1-4. [4] estimators � y ðUÞ ðS 1 Þ in Populations 1-3 and [1] estimator � y ðUÞ ðHRÞ in Population 4 show the poor performance but the proposed unbiased estimator � y ðUÞ ðPÞ have an excellent performance as compared to all considered estimators in all four populations 1-4.